Single-particle nondegeneracy and SU(3) fermion dynamical symmetry.

It is shown that the SU(3) symmetry of the fermion dynamical symmetry model is essentially preserved even for highly nondegenerate spherical single-particle energies. The breaking of SU(3) symmetry by single-particle energy terms for either normal deformation or superdeformation occurs only through an indirect Pauli effect and is significant only when the spherical single-particle splitting within shells is artificially large relative to that observed experimentally. 1 The shell model is commonly accepted as the microscopic basis for nuclear structure, but its practical implementation in medium and heavy nuclei requires a severe truncation of the model space. In the late 1980’s, motivated by the phenomenological successes of the Interacting Boson Model [1] and building on the schematic fermion model of Ginocchio [2], we proposed a symmetry-dictated truncation scheme for the shell model termed the Fermion Dynamical Symmetry Model (FDSM) [3,4]. The symmetry limits of the model and perturbations around these limits have been explored extensively, and found to be consistent with a broad range of nuclear structure observations [5]. It is now of interest to examine in detail the excursions from the symmetry limits of the theory, in order to test its suitability as a systematic truncation procedure for quantitative shell model calculations in heavy nuclei. Although symmetry breaking has been investigated in some cases, there is as yet no systematic analysis of such terms in the theory. In this paper, we initiate such an analysis for the Sp(6) ⊃ SU(3) limit of the FDSM. The Sp(6) ⊃ SU(3) dynamical symmetry of the FDSM may be identified with axially-symmetric rotational motion, and matrix elements derived in the symmetry limit or in perturbation around this limit have been shown to be in quantitative agreement with a variety of collective observables in heavy rotational nuclei [5]. The FDSM uses a modified Ginocchio coupling scheme that decomposes the single-particle angular momenta j of the shell model into an integer part k and a half-integer part i such that j = k + i. For orbitals exhibiting an Sp(6) ⊃ SU(3) dynamical symmetry, k = 1. In the lighter nuclei, there is a single value of i within a shell. In heavier nuclei there are typically 2–3 values of i within the normal-parity orbitals of a major shell, and for the enlarged valence spaces characteristic of superdeformation there may be as many as 5 values of i within a supershell [6]. A Hamiltonian with an Sp(6) ⊃ SU(3) dynamical symmetry requires that single-particle energy terms corresponding to the same value of i be degenerate. Thus, the symmetry-limit Hamiltonian will generally exhibit a higher level of degeneracy than the realistic spherical single-particle shell model spectrum, and quantitative calculations must consider the effect on the symmetry-limit results of symmetry-breaking by single-particle energies. We emphasize that these remarks concern the splitting of the 2 single-particle spectrum for the spherical shell model. The additional splitting associated with quadrupole interactions (the algebraic analog of Nilsson splittings at finite deformation) is a separate issue that is handled in the FDSM through quadrupole–quadrupole coupling terms of the 2-body Hamiltonian. The major portion of these terms respects the symmetry [5], and the remainder may be incorporated numerically where needed [7]. In the discussion of the single-particle splitting, the question of the physically relevant scale for the phenomenon is important. As has been discussed extensively in Refs. [6,5,8], for collective properties of many-body systems a natural scale is set by the dominant correlation energies. The issue that must be addressed is not simply the size of the single-particle splitting, but its size relative to the correlation energy of the system and how much of that splitting breaks the relevant symmetry. In particular, no matter how large the single-particle splitting terms are, they will have no influence on the SU(3) properties such as moments of inertia if they commute with the invariants of SU(3). Furthermore, even if they do not commute, their influence will be greatly suppressed if correlation energies in the system produce large energy separations between irreducible representations of the dynamical symmetry. To begin, we rewrite the single-particle energy in terms of the standard FDSM k–i basis: